We employ direct numerical simulations with an advanced finite difference method combined with the phase field method [1]}. The bubbles are put at the lower (hot) plate with pinned contact lines so that they cannot move or detach from the plate. To focus on the effects of the Rayleigh number (dimensionless strength of the thermal driving) and the bubble geometry, we disregard mass exchange between the bubble and the liquid, and keep the bubbles at constant volume. The calculations are performed for various Rayleigh numbers and geometries, expressed through the relative area covered by the bubbles, bubble height, bubble number, and type of bubble distribution. To define the thermal BL thickness in the multiphase system, we extend the traditional temperature profile slope method, based on heat flux conservation. With this, we propose an equivalent single-phase system, for which we apply the Grossmann-Lohse (GL) theory [2]}, [3]}, [4]} to predict the heat transfer and the temperature profile for RB convection with bubbles attached to the hot plate. Without introducing any new free parameter, these predictions well agree with our numerical results. They work for both moderate and large \(\mbox{\textit {Pr}}\) , which indicates that they can be applied to predict both the mass transport in water electrolysis with bubbles attached to the electrode surface and to catalytic surfaces on which bubbles have formed.
